Freddy Saia

Research

My research interests lie in number theory and arithmetic geometry. In particular, I am interested in (torsion of) elliptic curves and low-dimensional abelian varieties, often in the case of extra endomorphisms and from the vantage of rational points on modular and Shimura curves.

Preprints

• CM points on Shimura curves via QM-equivariant isogeny volcanoes (computations), (preprint)

• CM elliptic curves: volcanoes, reality and applications, part II (preprint)
  with P. L. Clark

Publications

• The least degree of a CM point on a modular curve (computations), (pdf)
  with P. L. Clark, T. Genao, and P. Pollack
  Journal of the London Mathematical Society, 2022

• Chevalley-Warning at the boundary (pdf)
  with P. L. Clark and T. Genao
  Expositiones Mathematicae, 2021

• Classically integral quadratic forms excepting at most two values (arXiv)
  with M. Barowsky, W. Damron, A. Mejia, N. Schock, and K. Thompson
  Proceedings of the AMS, 2018

Conferences and Workshops

Contributed Talks

• A volcanic approach to CM points on Shimura curves (slides)
  SRNTC, Louisiana State University – March 2023

• A volcanic approach to CM points on Shimura curves (slides)
  JMM Special Session on Arithmetic Geometry Informed by Computation, Boston, MA – January 2023

• A volcanic approach to CM points on Shimura curves (slides)
  PANTS XXXV, University of South Carolina – December 2022

• A volcanic approach to CM points on Shimura curves
  Graduate Summer Conference, UGA – July 2022

• Fields of moduli, rationality, and fake elliptic curves
  Graduate Summer Conference, UGA – July 2021

• No elliptic curve has a Q-rational point of order 13
  Graduate Summer Conference, UGA – July 2020

• Triangular numbers and modular forms
  Graduate Summer Conference, UGA – July 2019

• An axiomatic approach to homology theory
  Graduate Summer Conference, UGA – July 2018

• Classically integral quadratic forms excepting at most two values (slides)
  with M. Barowsky, W. Damron, A. Mejia, and N. Schock
  Joint Math Meetings, Atlanta, GA – January 2017

Attendee

• Virtual Number Theory and Arithmetic Geometry (VaNTAGe)
  Virtual – Fall 2020, Summer 2021, Fall 2021

• PCMI Graduate Summer School on
  Number Theory Informed by Computation
  Virtual – July 2021

• Mathematics Teacher-Scholar Symposium (MaTSS)
  Virtual – May 2021

• Connecticut Number Theory Conference (CTNT)
  Virtual – June 2020

• Front Range Number Theory Day
  Virtual – April 2020

• Arithmetic Geometry is Online In Zoom, Everyone (AGONIZE)
  Virtual – March 2020

• Arizona Winter School - Nonabelian Chabauty
  University of Arizona – March 2020

• Modular Forms, Arithmetic, and Women in Mathematics (MAAIM)
  Emory University – Nov 2019

• Midwest Arithmetic Geometry and Number Theory Series (MAGNTS)
  Ohio State University – Oct 2019

• CMI-HIMR Summer School in Computational Number Theory
  University of Bristol – June 2019

• Connecticut Summer School in Number Theory and
  Conference on Elliptic Curves and Modular Forms (CTNT)
  University of Connecticut – May 2018

Seminar Talks

• A volcanic approach to CM points on Shimura curves (slides)
  Algebra & Number Theory Seminar, Emory University – March 2023

• On Hindry–Silverman’s Sur le nombre de points de torsion rationnels sur une courbe elliptique Short and Sweet Seminar (SASS), UGA – October 2022

• A volcanic approach to CM points on Shimura curves (slides)
  Number Theory Seminar, UGA – September 2022

• The Chebotarev density theorem
  Catalan’s Conjecture after Mihailescu (CCAM) Reading Seminar, UGA – November 2021

• Isogeny volcanoes
  Graduate Student Seminar, UGA – October 2021

• On Honda’s Isogeny classes of abelian varieties over finite fields
  Classical Readings in Arithmetic/Algebraic Geometry Seminar (CRAAG), UGA – July 2021

• What is a Shimura curve?
  Graduate Student Seminar, UGA – April 2021

• Locally algebraic representations
  Serre on Zoom (SeZoom) reading seminar, UGA – March 2021

• Galois action on CM points
  Zannier on Zoom (ZaZoom) reading seminar, UGA – October 2020

• Sporadic CM points on Modular Curves
  Graduate Student Seminar, UGA – April 2020

• Formalism of moduli problems
  Arithmetic Moduli of Elliptic Curves Reading Seminar, UGA – Nov 2019

• Introduction to Sage and Magma
  Street Math and Related Topics Seminar (SMARTS), UGA – Nov 2019

• Modular forms part 2
  Arithmetic Moduli of Elliptic Curves Reading Seminar, UGA – Aug 2019

• Introduction to modular forms
  Graduate Algebraic Geometry Seminar, UGA – March 2019